Sums of Mixed Independent Positive Random Variables: A Unified Framework

TL;DR

This paper introduces a unified framework for deriving exact, compact solutions for the distribution of sums of various independent positive random variables, including new models like the alpha-mu mixture distribution, applicable to diverse fading scenarios.

Contribution

It presents a novel, comprehensive framework that simplifies the derivation of PDFs and CDFs for sums of mixed independent positive RVs, covering a wide range of fading models and introducing the alpha-mu mixture distribution.

Findings

Derived new tractable expressions for fading distributions

Unified various fading models under a single framework

Enabled exact analysis of sums of diverse RVs

Abstract

This paper proposes a comprehensive and unprecedented framework that streamlines the derivation of exact, compact -- yet tractable -- solutions for the probability density function (PDF) and cumulative distribution function (CDF) of the sum of a broad spectrum of mixed independent positive random variables (RVs). To showcase the framework's potential and extensive applicability, we tackle the enduring challenge of obtaining these statistics for the sum of fading variates in an exact, manageable, and unified manner. Specifically, we derive novel, tractable expressions for the PDF and CDF of the sum of Gaussian-class and non-Gaussian-class fading distributions, thereby covering a plethora of conventional, generalized, and recently introduced fading models. The proposed framework accommodates independent and identically distributed (i.i.d.) sums, independent but not necessarily identically distributed (i.n.i.d.) sums, and mixed-type sums. Moreover, we introduce the strikingly novel $\alpha$-$\mu$ mixture distribution that unifies all Gaussian-class fading models.